388: Correlation between the Numbers of Two Types of Children When the Family Size Distribution is Zero-Truncated Negative Binomal

Abstract

Let family size N be a random variable formed from two types of children B and C (B + C = N). The correlation coefficient between B and C has been recently studied by Rao et al. [1973]. We consider this problem when the frequency of fertile childless families (N = 0) is not available or cannot be estimated. Assuming the distribution of N is zero-truncated negative binomial, a formula for the correlation coefficient is derived. Applying this formula to the data of Reed and Reed [1965] gives a correlation coefficient of 0.27. The corresponding empirical coefficient for non-zero size families calculated directly from the data is 0.24.

Tracing out the effects of demographic changes on the income distribution

“…Once the counterfactual number of children H ht h t 0 ð Þ is estimated, two microsimulation exercises are carried out by replacing this estimate in the denominator of Equation (1), 16 A more realistic assumption is that children follow a Negative Binomial distribution (see Rao et al [22], Hamdan [14], and Wooldridge [26]). However, we use the Poisson model for two main reasons: (1) as mentioned above, estimators are still consistent when the real distribution is Negative Binomial (Poisson quasi-Maximum-Likelihood estimators), and (2) for two-parent households (that represent around 80% of the total households in the sample) it is not possible to reject the null hypothesis that the distribution of children per household is Poisson versus a Negative Binomial (model NB2, following Cameron and Trivedi [10]).…”

The Zero-Truncated Symmetrical Bivariate Negative Binomial Distribution

“…(2) Hamdan (1975) assumed that the family size Z has a zero-truncated negative binomial distribution and derived the formula for the correlation coefficient between X and Y as follows…”

“…Further, the formula for obtaining the correlation coefficient between X and Y in terms of the parameters when Z follows the zero-truncated negative binomial distribution is given by This formula may also be obtained from the formula derived by Hamdan ( 1975) after…”

“…They encountered the problem of isolating the sterile group from the total number of childless families to estimate the number of fertile childless families . Keeping this fact in view Hamdan (1975) derived a formula for the same when the frequency of fertile childless families (Z = 0) is either not available or cannot be determined. It is based on the assumption that the family size (Z) follows a zero-truncated negative binomial distribution.…”

Estructura demográfica e ingresos

“…En este caso F t|Zht (.) es la función de probabilidad acumulada de una variable con distribución Poisson con 34 Un supuesto algo más realista, y que se ha utilizado en la literatura, es que los hijos siguen una distribución binomial negativa -ver por ejemplo, Rao, Mazumdar, Waller y Li (1973), Hamdan (1975), y Wooldridge (2000). Sin embargo, en el presente trabajo se emplea el modelo de Poisson por dos motivos: 1) como ya se dijo más arriba, los estimadores siguen siendo consistentes y 2) para los hogares completos -que representan, dependiendo el año, entre el 63% y el 82% del total de hogares en la muestra-no puede rechazarse la hipótesis nula que la distribución de hijos por hogar es Poisson versus una Binomial Negativa -tipo NB2-sobre la base de un test de razón de verosimilitud.…”